Stein's method and a quantitative Lindeberg CLT for the Fourier transforms of random vectors

TL;DR

This paper develops a multivariate Stein's method approach to establish a quantitative Lindeberg CLT for the Fourier transforms of random vectors, providing explicit integral representations for the Hessian of the Stein equation solution.

Contribution

It introduces a novel multivariate Stein's method framework tailored for Fourier transforms of random vectors, with explicit Hessian integral representations.

Findings

Quantitative CLT for Fourier transforms of random vectors.

Explicit integral representation for the Hessian matrix.

Enhanced understanding of convergence rates in multivariate CLTs.

Abstract

We use a multivariate version of Stein's method to establish a quantitative Lindeberg CLT for the Fourier transforms of random $N$-vectors. We achieve this by deducing a specific integral representation for the Hessian matrix of a solution to the Stein equation with test function $e_t(x) = \exp(- i \sum_{k=1}^N t_k x_k)$, where $t,x \in \mathbb{R}^N$.